Integrand size = 29, antiderivative size = 70 \[ \int \frac {\left (d-e x^2\right )^{7/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\frac {2 x \left (d-e x^2\right )^{3/2}}{3 \left (d^2-e^2 x^4\right )^{3/2}}+\frac {x \sqrt {d-e x^2}}{3 d \sqrt {d^2-e^2 x^4}} \] Output:
2/3*x*(-e*x^2+d)^(3/2)/(-e^2*x^4+d^2)^(3/2)+1/3*x*(-e*x^2+d)^(1/2)/d/(-e^2 *x^4+d^2)^(1/2)
Time = 5.16 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.77 \[ \int \frac {\left (d-e x^2\right )^{7/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\frac {x \left (3 d+e x^2\right ) \sqrt {d^2-e^2 x^4}}{3 d \sqrt {d-e x^2} \left (d+e x^2\right )^2} \] Input:
Integrate[(d - e*x^2)^(7/2)/(d^2 - e^2*x^4)^(5/2),x]
Output:
(x*(3*d + e*x^2)*Sqrt[d^2 - e^2*x^4])/(3*d*Sqrt[d - e*x^2]*(d + e*x^2)^2)
Time = 0.33 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.24, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {1396, 292, 208}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (d-e x^2\right )^{7/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 1396 |
\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \int \frac {d-e x^2}{\left (e x^2+d\right )^{5/2}}dx}{\sqrt {d^2-e^2 x^4}}\) |
\(\Big \downarrow \) 292 |
\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {2}{3} \int \frac {1}{\left (e x^2+d\right )^{3/2}}dx+\frac {x \left (d-e x^2\right )}{3 d \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
\(\Big \downarrow \) 208 |
\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {2 x}{3 d \sqrt {d+e x^2}}+\frac {x \left (d-e x^2\right )}{3 d \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
Input:
Int[(d - e*x^2)^(7/2)/(d^2 - e^2*x^4)^(5/2),x]
Output:
(Sqrt[d - e*x^2]*Sqrt[d + e*x^2]*((x*(d - e*x^2))/(3*d*(d + e*x^2)^(3/2)) + (2*x)/(3*d*Sqrt[d + e*x^2])))/Sqrt[d^2 - e^2*x^4]
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> Si mp[(-x)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2*a*(p + 1))), x] - Simp[c*(q/( a*(p + 1))) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1), x], x] /; FreeQ[ {a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && EqQ[2*(p + q + 1) + 1, 0] && Gt Q[q, 0] && NeQ[p, -1]
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_)*((d_) + (e_.)*(x_)^(n_))^(q_.), x _Symbol] :> Simp[(a + c*x^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d + c*(x^n/e))^FracPart[p]) Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x], x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a* e^2, 0] && !IntegerQ[p] && !(EqQ[q, 1] && EqQ[n, 2])
Time = 0.22 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.67
method | result | size |
gosper | \(\frac {\left (e \,x^{2}+d \right ) x \left (e \,x^{2}+3 d \right ) \left (-e \,x^{2}+d \right )^{\frac {5}{2}}}{3 d \left (-e^{2} x^{4}+d^{2}\right )^{\frac {5}{2}}}\) | \(47\) |
orering | \(\frac {\left (e \,x^{2}+d \right ) x \left (e \,x^{2}+3 d \right ) \left (-e \,x^{2}+d \right )^{\frac {5}{2}}}{3 d \left (-e^{2} x^{4}+d^{2}\right )^{\frac {5}{2}}}\) | \(47\) |
default | \(\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, x \left (e \,x^{2}+3 d \right )}{3 \sqrt {-e \,x^{2}+d}\, d \left (e \,x^{2}+d \right )^{2}}\) | \(49\) |
Input:
int((-e*x^2+d)^(7/2)/(-e^2*x^4+d^2)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/3*(e*x^2+d)*x*(e*x^2+3*d)*(-e*x^2+d)^(5/2)/d/(-e^2*x^4+d^2)^(5/2)
Time = 0.07 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.01 \[ \int \frac {\left (d-e x^2\right )^{7/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=-\frac {\sqrt {-e^{2} x^{4} + d^{2}} {\left (e x^{3} + 3 \, d x\right )} \sqrt {-e x^{2} + d}}{3 \, {\left (d e^{3} x^{6} + d^{2} e^{2} x^{4} - d^{3} e x^{2} - d^{4}\right )}} \] Input:
integrate((-e*x^2+d)^(7/2)/(-e^2*x^4+d^2)^(5/2),x, algorithm="fricas")
Output:
-1/3*sqrt(-e^2*x^4 + d^2)*(e*x^3 + 3*d*x)*sqrt(-e*x^2 + d)/(d*e^3*x^6 + d^ 2*e^2*x^4 - d^3*e*x^2 - d^4)
\[ \int \frac {\left (d-e x^2\right )^{7/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\int \frac {\left (d - e x^{2}\right )^{\frac {7}{2}}}{\left (- \left (- d + e x^{2}\right ) \left (d + e x^{2}\right )\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((-e*x**2+d)**(7/2)/(-e**2*x**4+d**2)**(5/2),x)
Output:
Integral((d - e*x**2)**(7/2)/(-(-d + e*x**2)*(d + e*x**2))**(5/2), x)
\[ \int \frac {\left (d-e x^2\right )^{7/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\int { \frac {{\left (-e x^{2} + d\right )}^{\frac {7}{2}}}{{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((-e*x^2+d)^(7/2)/(-e^2*x^4+d^2)^(5/2),x, algorithm="maxima")
Output:
integrate((-e*x^2 + d)^(7/2)/(-e^2*x^4 + d^2)^(5/2), x)
\[ \int \frac {\left (d-e x^2\right )^{7/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\int { \frac {{\left (-e x^{2} + d\right )}^{\frac {7}{2}}}{{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((-e*x^2+d)^(7/2)/(-e^2*x^4+d^2)^(5/2),x, algorithm="giac")
Output:
integrate((-e*x^2 + d)^(7/2)/(-e^2*x^4 + d^2)^(5/2), x)
Time = 17.33 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.23 \[ \int \frac {\left (d-e x^2\right )^{7/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=-\frac {\sqrt {d^2-e^2\,x^4}\,\left (\frac {x\,\sqrt {d-e\,x^2}}{e^3}+\frac {x^3\,\sqrt {d-e\,x^2}}{3\,d\,e^2}\right )}{x^6-\frac {d^3}{e^3}+\frac {d\,x^4}{e}-\frac {d^2\,x^2}{e^2}} \] Input:
int((d - e*x^2)^(7/2)/(d^2 - e^2*x^4)^(5/2),x)
Output:
-((d^2 - e^2*x^4)^(1/2)*((x*(d - e*x^2)^(1/2))/e^3 + (x^3*(d - e*x^2)^(1/2 ))/(3*d*e^2)))/(x^6 - d^3/e^3 + (d*x^4)/e - (d^2*x^2)/e^2)
Time = 0.20 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.19 \[ \int \frac {\left (d-e x^2\right )^{7/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\frac {3 \sqrt {e \,x^{2}+d}\, d e x +\sqrt {e \,x^{2}+d}\, e^{2} x^{3}-3 \sqrt {e}\, d^{2}-6 \sqrt {e}\, d e \,x^{2}-3 \sqrt {e}\, e^{2} x^{4}}{3 d e \left (e^{2} x^{4}+2 d e \,x^{2}+d^{2}\right )} \] Input:
int((-e*x^2+d)^(7/2)/(-e^2*x^4+d^2)^(5/2),x)
Output:
(3*sqrt(d + e*x**2)*d*e*x + sqrt(d + e*x**2)*e**2*x**3 - 3*sqrt(e)*d**2 - 6*sqrt(e)*d*e*x**2 - 3*sqrt(e)*e**2*x**4)/(3*d*e*(d**2 + 2*d*e*x**2 + e**2 *x**4))